- Email: [email protected]
Comments on “Dissipative analysis and control of state-space symmetric systems” [Automatica 45 (2009) 1574–1579]
Automatica 48 (2012) 2734–2736
Contents lists available at SciVerse ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Correspondence
Comments on ‘‘Dissipative analysis and control of state-space symmetric systems’’ [Automatica 45 (2009) 1574–1579]✩ Gabriela Iuliana Bara 1 University of Strasbourg, LSIIT-UMR CNRS 7005, bd. Sébastien Brant, BP 10413, 67412 Illkirch, France
article
abstract
info
Article history: Received 11 April 2011 Received in revised form 23 November 2011 Accepted 20 March 2012 Available online 12 July 2012
We show that some results presented by Meisami-Azad, Mohammadpour, and Grigoriadis [MeisamiAzad, M., Mohammadpour, J. & Grigoriadis, K. M. (2009). Dissipative analysis and control of state-space symmetric systems. Automatica, 45, 1574–1579] are erroneous. First, we present a counterexample contradicting the necessary and sufficient condition of Lemma 5. Then, we point out that Lemma 4, which is a key instrumental lemma in the proof of Theorem 6, is also erroneous and we propose a new lemma that corrects and completes the proof of Theorem 6. Finally, we show that H∞ norm formulations claimed by Theorems 6, 7 and 8 are inaccurate which is also sustained by previously published results as well as numerical examples taken from the commented paper. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Linear systems State-space symmetric systems Dissipativity analysis
1. Introduction In this correspondence, a review of quadratic dissipativity analysis and control results proposed in Meisami-Azad, Mohammadpour, and Grigoriadis (2009) for internally symmetric systems is presented. We adopt the same notation as that in Meisami-Azad et al. (2009). (·⋆) stands for equations in (Meisami-Azad et al., 2009) while (·) designates equations in this contribution. 2. A counterexample for Lemma 5 in Meisami-Azad et al. (2009) Based on Theorem 1 in Meisami-Azad et al. (2009), the necessary and sufficient condition (14⋆) for quadratic dissipativity analysis of symmetric systems has been presented in Lemma 5 in Meisami-Azad et al. (2009). Unfortunately, the necessity of (14⋆) is incorrect, as shown by the following counterexample. Consider the state-space matrices A = C =
−
0.1 0.2
1 0
0
1
0.2 0.6
0 0
−0.5647 0.4639 −0.0100
0.4639 −2.5712 −0.3079
−0.0100 −0.3079 , BT −0.6385
=
, D = 02×2 , and the weighting matrices Q =
, S = 02×2 , and R =
1 0.4
0.4 5
. The system is asymp-
totically stable and, based on Theorem 1 in Meisami-Azad et al.
(2009), it is also strictly (Q , S , R)-dissipative. The storage function guaranteeing the asymptotic stability and the strict dissipativ ity is V (x(t )) = x(t )T Px(t ), with P =
recommended for publication in revised form by Associate Editor Lihua Xie under the direction of Editor André L. Tits. E-mail address: [email protected]. 1 Tel.: +33 368854862; fax: +33 368854480. 0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.06.015
−0.9688 6.5397 0.4759
2.3329 0.4759 27.8181
.
Linear matrix inequality (LMI) condition (14⋆) of Lemma 5 in Meisami-Azad et al. (2009) has been found infeasible. This proves that (14⋆) is only a sufficient condition for strict (Q , S , R)dissipativity (its feasibility ensures a quadratic storage function with P = α I) and that it is not a necessary one. This inaccuracy is due to the fact that the necessity proof has been delegated, for brevity reasons, to the proof of Theorem 6 in Meisami-Azad et al. (2009), which cannot be used since it deals with a particular case of (Q , S , R)-dissipativity. Moreover, the proof of Theorem 6 relies on some incorrect arguments, as shown in the following. 3. Proof of Theorem 6 in Meisami-Azad et al. (2009) Theorem 6 of Meisami-Azad et al. (2009) claims to present an explicit formula for computing the H∞ norm of symmetric systems. The proof of this theorem relies on Lemma 4 in MeisamiAzad et al. (2009), which is erroneous, as shown by the following counterexample. Consider i = 1 in Lemma 4 in MeisamiAzad et al. (2009) and A1 =
0.4 ✩ The material in this paper was not presented at any conference. This paper was
0.8997 −0.9688 2.3329
0.2 −0.3
−0.5
−0.1 . The matrix P 0.2
0.1746 0.0330
=
−0.2051 0.0740 0.0053 −0.0425 and B1 = 1.8 −0.44 1.8 5 0.2 > 0 and its −0.44 0.2 0.2
2.04
inverse are both solutions of inequality (13⋆). However, the expression A1 B1 + BT1 AT1 is not negative definite, since its eigenvalues are {−0.0596, 0.0218}. This counterexample shows that Lemma 4 in Meisami-Azad et al. (2009) is incorrect. This inconsistency is due
G.I. Bara / Automatica 48 (2012) 2734–2736
to a flaw in the proof of Lemma 4, which is explained in the following. Assuming that both P0 and P0−1 are solutions of inequality (13⋆), it follows, as correctly stated in Meisami-Azad et al. (2009), that P1 > 0 satisfies (13⋆) because of the convexity of the latter. However, we can not deduce that P1−1 also satisfies the same condition, as erroneously stated in Meisami-Azad et al. (2009). In fact, since P0 and P0−1 satisfy (13⋆), it follows that the convex combination of (13⋆) with respect to P0 and (13⋆) with respect to P0−1 is also true, which means that P1 is a solution of (13⋆). But, this procedure cannot be applied for P1−1 = UDiag(1, 1/σ¯ 2 , . . . , 1/σ¯ n )U T . This is because 1/σ¯ i for i = 2, . . . , n – with σ¯ i a convex combination of σi and σi−1 for i = 2, . . . , n – are not necessarily convex combinations of 1/σi and 1/σi−1 = σi . In other words, in general, the inverse of a convex combination of several matrices is not necessarily a convex combination of the matrices or their inverses. Therefore, P1−1 is not necessarily a solution of (13⋆). The following lemma should be employed instead of Lemma 4 in the proof of Theorem 6 of Meisami-Azad et al. (2009).
2735
2.5 2.0833
1
0
0
0.2
0.4
0.6
0.8
1
Fig. 1. Example 1 from Meisami-Azad et al. (2009).
Lemma 3.1. Consider the following quadratic matrix inequality with respect to a symmetric matrix P:
give only the optimal (minimum) value of γ for which system (1⋆) satisfies the mixed H∞ and positive real performance. The equality between this optimal value γ¯ , given either by (15⋆) or (22⋆), and the H∞ norm holds only when θ = 1. Indeed, it has been shown in Tan and Grigoriadis (2001) that the H∞ norm of system (1⋆) depends only on the system state-space matrices, since it is given by
ΩΩ T + P Ξ + Ξ P + P ΩΩ T P < 0,
γH∞ = max{λmax (−D), λmax (D − BT A−1 B)}.
(1)
where the symmetric matrix Ξ and the general matrix Ω are given. Then, there exists a symmetric positive-definite matrix P0 solution to inequality (1) if and only if ΩΩ T + Ξ < 0. Proof (Sufficiency). This is straightforward by choosing the solution P0 = I in (1). Necessity. Assume that P0 > 0 is a solution of (1). Pre- and postmultiplying (1) by P0−1 , we obtain the same inequality (1) with respect to P0−1 . Therefore, P0 and P0−1 are both solutions of (1). By the Schur complement, (1) is equivalent to
ΩΩ T + P0 Ξ + Ξ P0 Ω T P0
P0 Ω −I
< 0.
(2)
Now, the proof mainly follows the procedure employed in the proof of Lemma 2 in Tan and Grigoriadis (2001). Considering the eigenvalue decomposition of P0 = U ∆0 U T , where U T = U −1 and 1 T ∆0 = Diag(σ1 , . . . , σn ) > 0, then P0−1 = U ∆− 0 U . As σ1 > 0, there exists 0 ≤ λ1 ≤ 1 such that λ1 σ1 + (1 − λ1 )σ1−1 = 1. Therefore, the convex combination of (2) with respect to P0 and (2) with respect to P0−1 provides
ΩΩ T + P1 Ξ + Ξ P1 Ω T P1
P1 Ω −I
< 0,
(3)
where P1 = λ1 P0 + (1 − λ1 )P0−1 = UDiag(1, σ¯ 2 , . . . , σ¯ n )U T with σ¯ i = λ1 σi + (1 − λ1 )σi−1 for i = 2, . . . , n. Using the Schur complement on (3), and then pre- and post-multiplying the result by P1−1 , we obtain by the Schur complement the inequality (3), this time in P1−1 . Hence, P1 and P1−1 are both solutions of (3). By repeating the same procedure for the remaining eigenvalues, we obtain that Pn = UU T = I is a solution of (1). 4. Comments on H∞ norm formulations provided by Theorems 6, 7, and 8 in Meisami-Azad et al. (2009) When the trade-off parameter θ ∈ (0, 1) between H∞ and the positive real performance is given, it has been claimed in Theorems 6 and 7 in Meisami-Azad et al. (2009) that the H∞ norm of the stable symmetric system (1⋆) is given by relations (15⋆) and (22⋆), respectively. Note that neither of expressions (15⋆) and (22⋆) represents the H∞ norm of the system (1⋆). These expressions
(4)
Expressions (15⋆) and (22⋆) additionally depend on the tradeoff parameter θ . Therefore, the H∞ norm of a stable system is a constant value for a given system, and it cannot depend on the trade-off parameter θ , which can arbitrarily be chosen in the interval (0, 1). By analogy with the H∞ norm definition, the optimal value provided by (15⋆) and (22⋆) may be seen as the optimal mixed H∞ /PR (positive real) performance level. The following relationship between optimal value γ¯ and the H∞ norm can easily been shown. Lemma 4.1. If D ≥ 0, then the H∞ norm of the stable symmetric system (1⋆) and its optimal mixed H∞ /PR performance level γ¯ , defined by (22⋆), are connected through the relation γ¯ = f (θ )γH∞ . Proof. When D ≥ 0, it follows from formulation (4) given in Tan and Grigoriadis (2001) that the H∞ norm is γH∞ = λmax (D − BT A−1 B). In addition, relation (22⋆) reduces to γ¯ = λmax (−f (θ )BT A−1 B + (2 − θ2 + f (θ )−1 )D). Since 2 θ−θ 1 + f (θ )−1 = f (θ ), we deduce the relation of Lemma 4.1. Based on our analysis, γbound given by (30⋆) in Theorem 8 in Meisami-Azad et al. (2009) does not represent the optimally achievable level of H∞ performance by static output feedback (SOF) control as claimed in Meisami-Azad et al. (2009) but the optimally achievable level of mixed H∞ /PR performance by SOF control. Since D11 = 0, based on our Lemma 4.1, it follows that the optimally achievable level of H∞ performance by SOF control is independent of the trade-off parameter θ , and is given by T ⊥ ⊥T −1 ⊥ γH ∞/CL = λmax (BT1 B⊥ B2 B1 ). 2 (−B2 AB2 )
(5)
Moreover, for any γ > γbound , a symmetric SOF gain selected using (31⋆) renders the closed-loop system stable, but with a level of mixed H∞ /PR performance less than γ . By writing γ = αγbound with α ≥ 1, the selected gain ensures that the H∞ norm of the closed-loop system is less than αγH ∞/CL , where γH ∞/CL is given by (5). Note that the right-hand side of relation (31⋆) is independent of θ , since Σ is A + αγ 1 B1 BT1 , which confirms that the H∞ H ∞ /CL
norm of the closed-loop system is independent of the trade-off parameter θ . Let us consider Example 1 from Meisami-Azad et al. (2009). The computation of the H∞ norm using either the ‘norm’ Matlab function or expression (4) of Tan and Grigoriadis (2001) provided the same value 2.0833. Fig. 1 shows the optimal value γ¯ (blue
2736
G.I. Bara / Automatica 48 (2012) 2734–2736
2.5 2.168
1
0
0
0.2
0.4
0.6
0.8
1
a symmetric SOF control gain has been computed using (31⋆). Then, the H∞ norm of the closed-loop system has been computed using the ‘norm’ Matlab function for each θ . Fig. 2 shows the desired value (blue continuous line) and the actual value (green dashed–dotted line) of the closed-loop level γ as well as the H∞ norm of the closed-loop system (red dashed line), which is constant for all θ and is equal to 2.1680. This confirms that the H∞ norm of the closed-loop system is given by 1.01γH ∞/CL , where γH ∞/CL is 2.1465 by (5), as stated in our previous paragraph. Note that our Figs. 1 and 2 correct Figures 1 and 2 in Meisami-Azad et al. (2009). 5. Conclusion
Fig. 2. Example 2 from Meisami-Azad et al. (2009).
continuous line) computed using (22⋆) and γ¯ /f (θ ) (red dashed line) for different values of θ ∈ (0, 1). For this example, γ¯ /f (θ ) is constant, and is equal to the H∞ norm of the system. This is in agreement with our Lemma 4.1, since system matrix D is positive definite (D = 1). Consider now Example 2 from Meisami-Azad et al. (2009). By varying θ in the interval (0, 1), the optimal value γbound has been computed using (30⋆). By choosing the desired closed-loop level of mixed H∞ /PR performance as γ = 1.01γbound ,
In this correspondence, errors in Meisami-Azad et al. (2009) have been pointed out and corrections have been proposed. References Meisami-Azad, M., Mohammadpour, J., & Grigoriadis, K. M. (2009). Dissipative analysis and control of state-space symmetric systems. Automatica, 45, 1574–1579. Tan, K., & Grigoriadis, K. M. (2001). Stabilization and H∞ control of symmetric systems: an explicit solution. Systems & Control Letters, 44, 57–72.
Contents lists available at SciVerse ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Correspondence
Comments on ‘‘Dissipative analysis and control of state-space symmetric systems’’ [Automatica 45 (2009) 1574–1579]✩ Gabriela Iuliana Bara 1 University of Strasbourg, LSIIT-UMR CNRS 7005, bd. Sébastien Brant, BP 10413, 67412 Illkirch, France
article
abstract
info
Article history: Received 11 April 2011 Received in revised form 23 November 2011 Accepted 20 March 2012 Available online 12 July 2012
We show that some results presented by Meisami-Azad, Mohammadpour, and Grigoriadis [MeisamiAzad, M., Mohammadpour, J. & Grigoriadis, K. M. (2009). Dissipative analysis and control of state-space symmetric systems. Automatica, 45, 1574–1579] are erroneous. First, we present a counterexample contradicting the necessary and sufficient condition of Lemma 5. Then, we point out that Lemma 4, which is a key instrumental lemma in the proof of Theorem 6, is also erroneous and we propose a new lemma that corrects and completes the proof of Theorem 6. Finally, we show that H∞ norm formulations claimed by Theorems 6, 7 and 8 are inaccurate which is also sustained by previously published results as well as numerical examples taken from the commented paper. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Linear systems State-space symmetric systems Dissipativity analysis
1. Introduction In this correspondence, a review of quadratic dissipativity analysis and control results proposed in Meisami-Azad, Mohammadpour, and Grigoriadis (2009) for internally symmetric systems is presented. We adopt the same notation as that in Meisami-Azad et al. (2009). (·⋆) stands for equations in (Meisami-Azad et al., 2009) while (·) designates equations in this contribution. 2. A counterexample for Lemma 5 in Meisami-Azad et al. (2009) Based on Theorem 1 in Meisami-Azad et al. (2009), the necessary and sufficient condition (14⋆) for quadratic dissipativity analysis of symmetric systems has been presented in Lemma 5 in Meisami-Azad et al. (2009). Unfortunately, the necessity of (14⋆) is incorrect, as shown by the following counterexample. Consider the state-space matrices A = C =
−
0.1 0.2
1 0
0
1
0.2 0.6
0 0
−0.5647 0.4639 −0.0100
0.4639 −2.5712 −0.3079
−0.0100 −0.3079 , BT −0.6385
=
, D = 02×2 , and the weighting matrices Q =
, S = 02×2 , and R =
1 0.4
0.4 5
. The system is asymp-
totically stable and, based on Theorem 1 in Meisami-Azad et al.
(2009), it is also strictly (Q , S , R)-dissipative. The storage function guaranteeing the asymptotic stability and the strict dissipativ ity is V (x(t )) = x(t )T Px(t ), with P =
recommended for publication in revised form by Associate Editor Lihua Xie under the direction of Editor André L. Tits. E-mail address: [email protected]. 1 Tel.: +33 368854862; fax: +33 368854480. 0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.06.015
−0.9688 6.5397 0.4759
2.3329 0.4759 27.8181
.
Linear matrix inequality (LMI) condition (14⋆) of Lemma 5 in Meisami-Azad et al. (2009) has been found infeasible. This proves that (14⋆) is only a sufficient condition for strict (Q , S , R)dissipativity (its feasibility ensures a quadratic storage function with P = α I) and that it is not a necessary one. This inaccuracy is due to the fact that the necessity proof has been delegated, for brevity reasons, to the proof of Theorem 6 in Meisami-Azad et al. (2009), which cannot be used since it deals with a particular case of (Q , S , R)-dissipativity. Moreover, the proof of Theorem 6 relies on some incorrect arguments, as shown in the following. 3. Proof of Theorem 6 in Meisami-Azad et al. (2009) Theorem 6 of Meisami-Azad et al. (2009) claims to present an explicit formula for computing the H∞ norm of symmetric systems. The proof of this theorem relies on Lemma 4 in MeisamiAzad et al. (2009), which is erroneous, as shown by the following counterexample. Consider i = 1 in Lemma 4 in MeisamiAzad et al. (2009) and A1 =
0.4 ✩ The material in this paper was not presented at any conference. This paper was
0.8997 −0.9688 2.3329
0.2 −0.3
−0.5
−0.1 . The matrix P 0.2
0.1746 0.0330
=
−0.2051 0.0740 0.0053 −0.0425 and B1 = 1.8 −0.44 1.8 5 0.2 > 0 and its −0.44 0.2 0.2
2.04
inverse are both solutions of inequality (13⋆). However, the expression A1 B1 + BT1 AT1 is not negative definite, since its eigenvalues are {−0.0596, 0.0218}. This counterexample shows that Lemma 4 in Meisami-Azad et al. (2009) is incorrect. This inconsistency is due
G.I. Bara / Automatica 48 (2012) 2734–2736
to a flaw in the proof of Lemma 4, which is explained in the following. Assuming that both P0 and P0−1 are solutions of inequality (13⋆), it follows, as correctly stated in Meisami-Azad et al. (2009), that P1 > 0 satisfies (13⋆) because of the convexity of the latter. However, we can not deduce that P1−1 also satisfies the same condition, as erroneously stated in Meisami-Azad et al. (2009). In fact, since P0 and P0−1 satisfy (13⋆), it follows that the convex combination of (13⋆) with respect to P0 and (13⋆) with respect to P0−1 is also true, which means that P1 is a solution of (13⋆). But, this procedure cannot be applied for P1−1 = UDiag(1, 1/σ¯ 2 , . . . , 1/σ¯ n )U T . This is because 1/σ¯ i for i = 2, . . . , n – with σ¯ i a convex combination of σi and σi−1 for i = 2, . . . , n – are not necessarily convex combinations of 1/σi and 1/σi−1 = σi . In other words, in general, the inverse of a convex combination of several matrices is not necessarily a convex combination of the matrices or their inverses. Therefore, P1−1 is not necessarily a solution of (13⋆). The following lemma should be employed instead of Lemma 4 in the proof of Theorem 6 of Meisami-Azad et al. (2009).
2735
2.5 2.0833
1
0
0
0.2
0.4
0.6
0.8
1
Fig. 1. Example 1 from Meisami-Azad et al. (2009).
Lemma 3.1. Consider the following quadratic matrix inequality with respect to a symmetric matrix P:
give only the optimal (minimum) value of γ for which system (1⋆) satisfies the mixed H∞ and positive real performance. The equality between this optimal value γ¯ , given either by (15⋆) or (22⋆), and the H∞ norm holds only when θ = 1. Indeed, it has been shown in Tan and Grigoriadis (2001) that the H∞ norm of system (1⋆) depends only on the system state-space matrices, since it is given by
ΩΩ T + P Ξ + Ξ P + P ΩΩ T P < 0,
γH∞ = max{λmax (−D), λmax (D − BT A−1 B)}.
(1)
where the symmetric matrix Ξ and the general matrix Ω are given. Then, there exists a symmetric positive-definite matrix P0 solution to inequality (1) if and only if ΩΩ T + Ξ < 0. Proof (Sufficiency). This is straightforward by choosing the solution P0 = I in (1). Necessity. Assume that P0 > 0 is a solution of (1). Pre- and postmultiplying (1) by P0−1 , we obtain the same inequality (1) with respect to P0−1 . Therefore, P0 and P0−1 are both solutions of (1). By the Schur complement, (1) is equivalent to
ΩΩ T + P0 Ξ + Ξ P0 Ω T P0
P0 Ω −I
< 0.
(2)
Now, the proof mainly follows the procedure employed in the proof of Lemma 2 in Tan and Grigoriadis (2001). Considering the eigenvalue decomposition of P0 = U ∆0 U T , where U T = U −1 and 1 T ∆0 = Diag(σ1 , . . . , σn ) > 0, then P0−1 = U ∆− 0 U . As σ1 > 0, there exists 0 ≤ λ1 ≤ 1 such that λ1 σ1 + (1 − λ1 )σ1−1 = 1. Therefore, the convex combination of (2) with respect to P0 and (2) with respect to P0−1 provides
ΩΩ T + P1 Ξ + Ξ P1 Ω T P1
P1 Ω −I
< 0,
(3)
where P1 = λ1 P0 + (1 − λ1 )P0−1 = UDiag(1, σ¯ 2 , . . . , σ¯ n )U T with σ¯ i = λ1 σi + (1 − λ1 )σi−1 for i = 2, . . . , n. Using the Schur complement on (3), and then pre- and post-multiplying the result by P1−1 , we obtain by the Schur complement the inequality (3), this time in P1−1 . Hence, P1 and P1−1 are both solutions of (3). By repeating the same procedure for the remaining eigenvalues, we obtain that Pn = UU T = I is a solution of (1). 4. Comments on H∞ norm formulations provided by Theorems 6, 7, and 8 in Meisami-Azad et al. (2009) When the trade-off parameter θ ∈ (0, 1) between H∞ and the positive real performance is given, it has been claimed in Theorems 6 and 7 in Meisami-Azad et al. (2009) that the H∞ norm of the stable symmetric system (1⋆) is given by relations (15⋆) and (22⋆), respectively. Note that neither of expressions (15⋆) and (22⋆) represents the H∞ norm of the system (1⋆). These expressions
(4)
Expressions (15⋆) and (22⋆) additionally depend on the tradeoff parameter θ . Therefore, the H∞ norm of a stable system is a constant value for a given system, and it cannot depend on the trade-off parameter θ , which can arbitrarily be chosen in the interval (0, 1). By analogy with the H∞ norm definition, the optimal value provided by (15⋆) and (22⋆) may be seen as the optimal mixed H∞ /PR (positive real) performance level. The following relationship between optimal value γ¯ and the H∞ norm can easily been shown. Lemma 4.1. If D ≥ 0, then the H∞ norm of the stable symmetric system (1⋆) and its optimal mixed H∞ /PR performance level γ¯ , defined by (22⋆), are connected through the relation γ¯ = f (θ )γH∞ . Proof. When D ≥ 0, it follows from formulation (4) given in Tan and Grigoriadis (2001) that the H∞ norm is γH∞ = λmax (D − BT A−1 B). In addition, relation (22⋆) reduces to γ¯ = λmax (−f (θ )BT A−1 B + (2 − θ2 + f (θ )−1 )D). Since 2 θ−θ 1 + f (θ )−1 = f (θ ), we deduce the relation of Lemma 4.1. Based on our analysis, γbound given by (30⋆) in Theorem 8 in Meisami-Azad et al. (2009) does not represent the optimally achievable level of H∞ performance by static output feedback (SOF) control as claimed in Meisami-Azad et al. (2009) but the optimally achievable level of mixed H∞ /PR performance by SOF control. Since D11 = 0, based on our Lemma 4.1, it follows that the optimally achievable level of H∞ performance by SOF control is independent of the trade-off parameter θ , and is given by T ⊥ ⊥T −1 ⊥ γH ∞/CL = λmax (BT1 B⊥ B2 B1 ). 2 (−B2 AB2 )
(5)
Moreover, for any γ > γbound , a symmetric SOF gain selected using (31⋆) renders the closed-loop system stable, but with a level of mixed H∞ /PR performance less than γ . By writing γ = αγbound with α ≥ 1, the selected gain ensures that the H∞ norm of the closed-loop system is less than αγH ∞/CL , where γH ∞/CL is given by (5). Note that the right-hand side of relation (31⋆) is independent of θ , since Σ is A + αγ 1 B1 BT1 , which confirms that the H∞ H ∞ /CL
norm of the closed-loop system is independent of the trade-off parameter θ . Let us consider Example 1 from Meisami-Azad et al. (2009). The computation of the H∞ norm using either the ‘norm’ Matlab function or expression (4) of Tan and Grigoriadis (2001) provided the same value 2.0833. Fig. 1 shows the optimal value γ¯ (blue
2736
G.I. Bara / Automatica 48 (2012) 2734–2736
2.5 2.168
1
0
0
0.2
0.4
0.6
0.8
1
a symmetric SOF control gain has been computed using (31⋆). Then, the H∞ norm of the closed-loop system has been computed using the ‘norm’ Matlab function for each θ . Fig. 2 shows the desired value (blue continuous line) and the actual value (green dashed–dotted line) of the closed-loop level γ as well as the H∞ norm of the closed-loop system (red dashed line), which is constant for all θ and is equal to 2.1680. This confirms that the H∞ norm of the closed-loop system is given by 1.01γH ∞/CL , where γH ∞/CL is 2.1465 by (5), as stated in our previous paragraph. Note that our Figs. 1 and 2 correct Figures 1 and 2 in Meisami-Azad et al. (2009). 5. Conclusion
Fig. 2. Example 2 from Meisami-Azad et al. (2009).
continuous line) computed using (22⋆) and γ¯ /f (θ ) (red dashed line) for different values of θ ∈ (0, 1). For this example, γ¯ /f (θ ) is constant, and is equal to the H∞ norm of the system. This is in agreement with our Lemma 4.1, since system matrix D is positive definite (D = 1). Consider now Example 2 from Meisami-Azad et al. (2009). By varying θ in the interval (0, 1), the optimal value γbound has been computed using (30⋆). By choosing the desired closed-loop level of mixed H∞ /PR performance as γ = 1.01γbound ,
In this correspondence, errors in Meisami-Azad et al. (2009) have been pointed out and corrections have been proposed. References Meisami-Azad, M., Mohammadpour, J., & Grigoriadis, K. M. (2009). Dissipative analysis and control of state-space symmetric systems. Automatica, 45, 1574–1579. Tan, K., & Grigoriadis, K. M. (2001). Stabilization and H∞ control of symmetric systems: an explicit solution. Systems & Control Letters, 44, 57–72.